Gamow-Teller transitions of neutron-rich $N=82,81$ nuclei by shell-model calculations

$\beta$-decay half-lives of neutron-rich nuclei around $N=82$ are key data to understand the $r$-process nucleosynthesis. We performed large-scale shell-model calculations in this region using a newly constructed shell-model Hamiltonian, and successfully described the low-lying spectra and half-lives of neutron-rich $N=82$ and $N=81$ isotones with $Z=42-49$ in a unified way. We found that their Gamow-Teller strength distributions have a peak in the low-excitation energies, which significantly contributes to the half-lives. This peak, dominated by $\nu 0g_{7\mathrm{/}2}\to \pi 0g_{9\mathrm{/}2}$ transitions, is enhanced on the proton deficient side because the Pauli-blocking effect caused by occupying the valence proton $0g_{9\mathrm{/}2}$ orbit is weakened.